Theorem sbn1 39151

Description: One direction of sbn 2287, using fewer axioms. Compare 19.2 1981. (Contributed by Steven Nguyen, 18-Aug-2023.)

Assertion

Ref	Expression
sbn1	⊢ ([𝑡 / 𝑥] ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑)

Proof of Theorem sbn1

Step	Hyp	Ref	Expression
1		fal 1551	. . 3 ⊢ ¬ ⊥
2	1	nsb 39150	. 2 ⊢ ¬ [𝑡 / 𝑥]⊥
3		pm2.21 123	. . 3 ⊢ (¬ 𝜑 → (𝜑 → ⊥))
4	3	sb2imi 2080	. 2 ⊢ ([𝑡 / 𝑥] ¬ 𝜑 → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]⊥))
5	2, 4	mtoi 201	1 ⊢ ([𝑡 / 𝑥] ¬ 𝜑 → ¬ [𝑡 / 𝑥]𝜑)

Syntax hints:  ¬ wn 3   → wi 4  ⊥wfal 1549  [wsb 2069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970

This theorem depends on definitions:  df-bi 209  df-tru 1540  df-fal 1550  df-ex 1781  df-sb 2070

This theorem is referenced by: (None)